Problem: Simplify the following expression and state the condition under which the simplification is valid: $z = \dfrac{a^2 - 5a - 14}{a^2 + 2a}$
First factor the expressions in the numerator and denominator. $ \dfrac{a^2 - 5a - 14}{a^2 + 2a} = \dfrac{(a - 7)(a + 2)}{(a)(a + 2)} $ Notice that the term $(a + 2)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(a + 2)$ gives: $z = \dfrac{a - 7}{a}$ Since we divided by $(a + 2)$, $a \neq -2$. $z = \dfrac{a - 7}{a}; \space a \neq -2$